Rotating lines of simultaneity Sending out a signal
Spacetime Diagrams II
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1
Spacetime Diagrams II
Spacetime diagram (or map)
Plots space on one axis and time on the other Payoffs
Place space and time on equal footing Review history of events and motions along a given line in spacetime Plot same events in different spacetime diagrams for different inertial reference frames Can find what is different and what is the same between the
frames same place
same time
D
time B
Reference event O, along with other events A, B, C, and D. The dashed horizontal line represents events simultaneous in time while the dashed vertical line represents events at the same location.
C
A O
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space
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Same events, different frames
Considers a light flash reflected off a mirror and back as seen from three different frames
rocket, lab, and super-rocket The mirrors are at rest in the rocket frame and the superrocket is traveling in the same direction but faster (relative to lab frame) than the rocket
The light paths look like that shown below: rocket
lab
super-rocket
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2
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Spacetime interval is conserved
The spacetime interval is conserved (the same for each frame). (interval)2 = (time)2 – (space)2 = constant
The red hyperbola in the spacetime plots show the line of constant interval. The arrows represent the same arrow in spacetime!
lab time
R
Maps show different perspectives of the same arrow in spacetime.
Spacetime geometry is Lorentz geometry, not Euclidean.
lab space
rocket time
super-rocket time R
R
super-rocket space
rocket space
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Worldline
A worldline tracks the passage of a particle through spacetime
The worldline is a path in spacetime
Every particle has a worldline The spacetime map is an image of the worldline of a particle in a particular inertial frame Clocks don’t move (in x) in lab frame but move along in time – time passes at lattice points so that worldline of clocks are vertical
In the lab frame the line of simultaneity is horizontal Note: All particles have worldlines, even “stationary” ones. 5
1
2
1
2
3
4 5
4
3 Worldline of particles in spacetime for the lab frame. The world line for particle 1 contains a sample set of event points that make up the worldline.
time
O
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Trajectories of particles in space (not spacetime). Each particle starts at the reference clock (square) and moves with constant velocity.
space
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3
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Non-constant speed particles
Accelerated particles
Have curved worldlines which represent their non-constant speed (acceleration is due to a force on the particle) This is true even if the particle is moving in a spatially straight line, like a train on a track.
Usefulness of spacetime diagrams –
Useful for recognizing patterns of events for looking at the laws of nature but useless for influencing the events they represent (since events are recorded and sent back to the (ideal) observer) limits on worldline slope
time
Worldline of an accelerated particle. The particle stops at Z and then continues to P.
P
Z At every point in spacetime the particle is limited by the speed of light which place a limiting slope in the spacetime diagram.
O
space
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Length along a path
Distance along a spatial path
We could use a flexible tape or short stick to measure spatial distance along a track or path (Spatial distance)2 = (x2 - x1)2 + (y2 - y1)2 for each segment Sum together distances to get the total length The path lengths are the same for all surveyors
Different paths give different lengths Straight line
Shortest path/distance between two points 14 10
The total length of the curved/winding path is long that the straight line between the two end points. The increased length for any segment is given by
12
8
10
6
8
north
increase increase increase in length in north in east 2
6
4 4
2 1/ 2
2 2 O
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east
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Proper time along a worldline
Wristwatch (proper) time
Sum spacetime interval along a worldline in Lorentz geometry the same way distance measures path length in Euclidean geometry (interval)2 = (t2 - t1)2 - (x2 - x1)2 for each segment Sum together the intervals to get the total proper time The proper time is the same for all inertial observers
Different world lines give different total proper times Straight worldline (free particle)
Longest proper time between two points 8
The total proper time of the curved/winding worldline is shorter than the proper time for the straight worldline between the two end points. The increased proper time for any segment is given by
10 8
6
time
2 increase in advance proper time in time
6 4
4 2
2
O
2 1/ 2
Principle of Maximal Aging: a free particle follows the worldline of maximal aging
space
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increment in space
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Principle of Maximal Aging
A free particle follows the worldline of maximal aging
In Special Relativity straight worldlines have the longest proper time between two points
All inertial observers agree on which worldline is straight and has the longest proper time
Free particles follow straight worldlines
Worldline in Spacetime
increase in east
north
direct path
O
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increase in space
time
increase in north
increase 2 in north
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B
Path in Space
B
2 increase in east
straight worldline 1/ 2
increase in time
increase 2 increase 2 in space in time
increase in north
east
O
Spacetime Diagrams II
In spacetime the curved worldline is transversed in shorter proper time.
1/ 2
increase in time
space
10
5
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“Kinked” Worldline For a curved worldline the clock associated with the particle is accelerating We take the limit in which we have a large acceleration over a short time
The worldline is straight except for a negligible part of the time when it changes direction
space = 0 m time = 10 m
B
(total proper time along OPB) = 10 meters of time
Zero proper time for light time
(proper time along leg OR)2 = (5 m)2 – (5 m)2 =0
space = 5 m time = 5 m Q
P
4
5
Reduced proper time along kinked worldline
space = 4 m time = 5 m
3
space = 0 m time = 0 m
(total proper time along leg OR)2 = 2x(proper time along OR ) =0
R
(proper time along leg OQ)2 = (5 m)2 – (4 m)2 = (3 m)2 (total proper time along leg OQB) = 2x(proper time along OQ ) =6m
4 O
space
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Sample Problem 5-1
Consider a traveler who moves along the solid straight lines in the figure below.
What is her proper time between the various events? What is here total proper time (aging) along the worldline 1, 2, 3, 4?
Her brother moves along the straight dotted line from event 1 to 4 directly.
Which twin is younger when they rejoin at event 4?
Calculate the increase in time on his clock 4
6 Event
Space (yr)
Time (yr)
1
1
0
2
1
1
3
-0.5
3
4
2
6
time
Kinked Path Event 1->2: (interval) = sqrt(12 – 02) = 1 yr 2->3: (interval) = sqrt(22 – 1.52) = 1.32 yr 3->4: (interval) = sqrt(32 – 2.52) = 1.66 yr
5 4
Total time:
3
Straight Path
2
Event 1->4: (interval) = sqrt(62 – 12)
2 1
-1
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0 1 space
= 5.92 yr
The brother (straight line path) ages more than the sister (kinked) path.
